Ion trapping

7/30/2019 Ion trapping

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Quantum information processing in ion traps IID. J. Wineland, NIST, Boulder

Part I,

Rainer Blatt

Lecture 1: Nuts and bolts

Ion trapology Qubits based on ground-state hyperfine levels Two-photon stimulated-Raman transitions

* Rabi rates, Stark shifts, spontaneous emission

Lecture 2: Quantum computation (QC) and quantum-limited measurement

Trapped-ion QC and DiVincenzos criteria

Gates Scaling Entanglement-enhanced quantum measurement

Lecture 3: Decoherence

Memory decoherence Decoherence during operations* technical fluctuations

* spontaneous emission

* scaling

Decoherence and the measurement problem


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Ion trapping 101

Earnshaws theorem: In a charge free region, cannot confine a charged particle

with static electric fields.Proof: For confinement, must have (2(q)/2xi)trap location < 0 (xi {x,y,z})But from Laplaces equation: 2 = 0, cannot satisfy confinement condition for all xi.

B0

U0

Solution 1: Penning trap:

q U0 [2z2 x2y2]

Difficult to accomplish individual

ion addressing.(However, see: Ciaramicoli, Marzoli,

Tombesi, PRL 91, 017901 (2003))

Solution 2: RF-Paul trap:

= (x2 + y2 + z2)V0

cost + U0

(x2 + y2 + z2)

[ + + = + + = 0] (Laplace)


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y

z

x

y x

axisztrap

end view

~2R

Special case:

linear RF trap(quadrupole mass

filter plugged on-axisWith

static fields)U

o

V0

cos T

t

Uc

(Get s and numerically)


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Equations of motion

(classical treatment adequate)Mathieu equation

(2)

z-motion, qz = 0 (static harmonic well)

x,y motion, Mathieu equation:

plug into (2), find (recursion relation for) C2n


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Solution in ith direction (i {x,y}):

|qi|

0 .2 .4 .6 .8 1.0

ai 0

-.2

.2

-.4

.4

.6

-.6

.8

1.0

stable

unstable

unstable

pseudo-potential strength

sta

tic

potential

strength


6/27|q |

0 .2 .4 .6 .8 1.0

ax 0

-.2

.2

-.4

.4

.6

-.6

.8

1.0

-.8-1.0

-1.2

ay0.2

-.2

.4

-.4

-.6

.6

-.8-1.0

.8

1.0

az

x = T/2

y = T/2

x 0

y 0

stable

ai Ui, ax + ay + az =0Simultaneous solution for x, y, z:


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0.05

typically, we live here

|ai| , qi2


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Heuristic approach: pseudo-potential approximation

assume mean ion position changes negligibly in duration 2/T

pseudo-potential from micromotion kinetic energy:

agrees with

Mathieu equation

limit |ai| , qi2


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Digression: Optical dipole traps:

outer

electron

L Response of atomic core tolaser field is negligible

because of heavy mass.

ForL >> 0 (blue detuning)electron response out of phase with

electric force. Electron trapped in

ponderomotive (pseudo-potential)

laser potential (just like RF trap).

Trapping in field minima.

Core is attached to electron

Dispersion:ForL


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(some) ion-trap realities

axisztrap

y x

EDC

Patch potentials:

Static potentials: pushes

ions away from trap axis

micromotion xsinTtcan cause X-tal heating

Fluctuating patch fields:

causes heating; COM

primarily affected

Source: unknown!(mobile electrons on

oxide layers,.. ??)


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Idealized trap:RF electrodes

control electrodes

1'

2'

3'

4'

1

2

3

4

IONS

200 mApproximation:

gold-coated

alumina wafers


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0.2 mm

Chris Myatt et al.

linear Paul (RF) trap

VRF

~ 500 V

RF ~ 50 250 MHz


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~ 1 cm


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0.2 mm

For9Be+, V0 = 500 V, T/2 = 200 MHz, R = 200 m

x,y/2 ~ 6 MHz


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Ion Trap QC: Proposal: J. I. Cirac and P. Zoller,PRL 74, 4091 (1995)

Motion data bus

(e.g., center-of-mass mode)Laser beam

n=3

n=2

n=1

n=0

Stay in two lowest

motional states (motion qubit)

Internal-state qubit


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n'

n

ladder states for selected motional mode

0

1

0

E - E =0

0

12

0 = optical frequency: single photon

need good laser frequency stability

memory and gate coherence limited by upper state lifetime (~ seconds)

0 = RF/microwave frequency memory coherence limited by upper-state lifetime (>> days)

sideband transitions weak at RF 2-photon optical stimulated-Raman transitions

frequency stability = RF modulator stability vary sideband coupling (Lamb-Dicke parameter) with k2 k1

gate decoherence: spontaneous-Raman scattering (fundamental limit)


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Stimulated-Raman transitions:

Simple case: Motion: 1-D harmonic well (frequency M

),

Internal states: 3-level system

rrk ,G

bbk ,G

M

0

e

,0

,1

,0

,1

e,0

e,1

,2

b = blue

r= red

electric dipole

transitions

e >> >> 0 >> M


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rrk ,G

bbk ,G

M

0

e

,0

,1

,0

,1

e,0

e,1

,2

b = blue

r= red

zero-point

wavefunction spread


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(+ rotating wave approximation)

similarly:


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Adiabatic elimination:

make ansatz:


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rrk ,G

bbk ,G

M

0

e

,0,1

,0

,1

e,0

e,1

,2

b = blue

r= red

Stark shift of from blue laser

Add in other Stark shifts


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Add in other Stark shifts

absorb Stark shifts into wave function amplitudes

near a resonance:

Rabi flopping


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(Lamb-Dicke parameter)

Be+: P = 1 mW, w0 = 25 m, /2 = 100 GHz, /2 ~ 0.5 MHz

Carrier transitions:

Debye-Waller factor

Sideband transitions: n = n 1(n> = larger of n and n)

red sideband (n = n-1): can get from

Jaynes-Cummings Hamiltonian from cavity-QED(see, e.g., Raimond, Brune, Haroche, Rev. Mod. Phys. 73, 565 (01))


25/27

Complete picture:

Sum over excited states, typically:

2P3/2

2S1/2

2P1/2

can tune out differential Stark shifts

can tune out polarization sensitivity

For N ions, consider effects of 3N modes

Debye-Waller factors from spectator modes

sideband transitions: interference from two-mode transitions:

e.g. np -mr= M


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Spontaneous emission:

rrk ,G

bbk ,G

0 ,1

,0

,1

,2

b = blue

r= red

e

e,1

e,0

e

,0

M


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Quantum information processing in ion traps IID. J. Wineland, NIST, Boulder

Lecture 1: Nuts and bolts

Ion trapology Qubits based on ground-state hyperfine levels

Two-photon stimulated-Raman transitions* Rabi rates, Stark shifts, spontaneous emission

Lecture 2: Quantum computation (QC) and quantum-limited measurement

Trapped-ion QC and DiVincenzos criteria

Gates Scaling Entanglement-enhanced quantum measurement

Lecture 3: Decoherence

Memory decoherence Decoherence during operations

* technical fluctuations

* spontaneous emission

* scaling

Decoherence and the measurement problem

Ion trapping

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